This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Hapa kuna suluhisho la maswali yako: Swali la 1: Pata mlinganyo wa mstari ulio wima kwa mstari y - 4x + 3 = 0 na unapita kwenye nukta (-8, 5). Step 1: Pata gradient ya mstari uliotolewa. Mlinganyo uliotolewa ni y - 4x + 3 = 0. Tunaweza kuandika upya kama y = 4x - 3. Gradient ya mstari huu, m_1, ni 4. Step 2: Pata gradient ya mstari ulio wima. Kwa mistari miwili iliyo wima, bidhaa ya gradient zao ni -1. m_1 · m_2 = -1 4 · m_2 = -1 m_2 = -(1)/(4) Gradient ya mstari ulio wima ni -(1)/(4). Step 3: Tumia gradient na nukta uliyopewa kupata mlinganyo wa mstari. Mstari ulio wima unapita kwenye nukta (-8, 5) na una gradient m_2 = -(1)/(4). Tunatumia fomu y - y_1 = m(x - x_1). y - 5 = -(1)/(4)(x - (-8)) y - 5 = -(1)/(4)(x + 8) Step 4: Rahisisha mlinganyo. Zidisha pande zote mbili kwa 4: 4(y - 5) = -(x + 8) 4y - 20 = -x - 8 Panga upya ili kupata fomu ax + by = c: x + 4y = 20 - 8 x + 4y = 12 Mlinganyo wa mstari ni x + 4y = 12. Swali la 2: Pata mlinganyo wa mstari ambao uko umbali sawa kutoka nukta (2, 3) na (6, 1), ukiandika katika fomu ax + by = c. Step 1: Pata nukta ya kati (midpoint) ya sehemu ya mstari inayounganisha nukta (2, 3) na (6, 1). Nukta ya kati M = ((x_1 + x_2)/(2), (y_1 + y_2)/(2)). M = ((2 + 6)/(2), (3 + 1)/(2)) M = ((8)/(2), (4)/(2)) M = (4, 2) Step 2: Pata gradient ya mstari unaounganisha nukta (2, 3) na (6, 1). Gradient m_1 = (y_2 - y_1)/(x_2 - x_1). m_1 = (1 - 3)/(6 - 2) m_1 = (-2)/(4) m_1 = -(1)/(2) Step 3: Pata gradient ya mstari ulio wima kwa mstari unaounganisha nukta hizo. Mstari unaotafutwa ni perpendicular bisector (mstari unaogawanya sehemu ya mstari kwa nusu na kuwa wima kwake). Gradient yake, m_2, itakuwa: m_1 · m_2 = -1 -(1)/(2) · m_2 = -1 m_2 = 2 Step 4: Tumia gradient m_2 na nukta ya kati M kupata mlinganyo wa mstari. Mstari unapita kwenye nukta (4, 2) na una gradient m_2 = 2. Tunatumia fomu y - y_1 = m(x - x_1). y - 2 = 2(x - 4) Step 5: Rahisisha mlinganyo katika fomu ax + by = c. y - 2 = 2x - 8 Panga upya: 2x - y = -2 + 8 2x - y = 6 Mlinganyo wa mstari ni 2x - y = 6. Tuma swali linalofuata 📸